Multivalued Functions : II

Visualization

By determining the geometries that complex functions reside in, it is possible to change the concept of a function from a particular, complicated kind of activity with singularities in its behavior, to a simple action in a unique space. Take for example, the Riemann surface for the square root, seen on the previous page. The two different values of the square root come naturally in the two sheets of the surface, and the branching at the origin is the natural effect of walking towards it in a straight line.

A second way of visualizing branch points and other singularities is to look at the characteristic action of the function, and then hypothesize a geometry appropriate to that action, one that might not immediately suggest itself. For example, the single infinite branch point of the complex logarithm (which encompasses the sine and cosine) was shown as its infinitely spiraling Riemann surface, but could also be seen as the simply periodic nature of motion along a cylinder. This technique, of finding the implied geometry of a process, by examining the way it acts, is known as analysis situs. Here you see the repeated motion of the sine of a circle transformed into cylindrical motion:

It has two locations of periodicity, around the two infinite branch points. The space lets the action be natural.

Two independent types of periodicity could be represented by motion on a torus, where there are two totally independent types of paths, similar to the single periodicity we saw on the cylinder. Here you see the double-periodicity, first as two discontinuous jumps on the plane, and then only one on a cylinder, and then no discontinuities on torus. The torus is the space in which the action is natural:

This periodicity could also be thought of as a periodic tiling of the plane, where as we move from one tile to the next, we have the same characteristics, but all of them become the same torus. On a clock, 1 o'clock, 13 o'clock, 25 o'clock, etc., all look the same: there are an infinity of numbers of hours that have the same appearance on the clock hands. But if there are two dimensions of periodicity, then there are a doubly infinite grid of answers that have the same appearance. If the function is periodic in two ways, then a particular location on one tile gives the same value as the equivalent position on any other tile. The equivalence gives this repeated tiling:

What if we have more such branch-points? What if we had not two, but three independent types of circuits that could be made?

Could this be represented as a plane tiling? Not in the usual sense. The surface on which it would have to exist would be something like a figure-eight, a double-holed torus. Here you have an animation where the simple plane is torn to form a cylinder, which is then joined to form a doubly periodic torus, which is then punctured to create a triply periodic double-holed torus. The basic torus has two independent types of periodicity: one around the whole loop, and one that circles the ring. The double-holed torus has three completely different ways of getting back to the same point: one along the loop of each torus, and one around the ring:

Another Type of Singularity

While branch-points are the most interesting singularity Riemann dealt with, there is one more type to discuss: poles. A pole is a location that acts as an infinity. Infinities come from an impossible ratio, as in 1/0. Just as 6/2 is the number of times 2 goes into 6, which is 3, 1/0 means the number of times 0 goes into 1. It doesn't matter how many zeroes you add together, you'll never get 1, making it a type of infinity. Because +0 and -0 are right next to each other, a pole jumps from positive infinity to negative infinity at a singular spot:

Just like branch-points, there are poles of different orders, depending on how infinite they are. For example 1/z2 would be a pole of second order, being doubly infinite. Here you see two poles being pulled together to become one. Nothing essential changes, and Riemann remarks that a pole of second order can be considered as two poles in the same position. Here are five poles coming together:

We can use analysis situs to figure out the geometry in which poles make sense. Rather than the complex plane, we can use what is now known as the “Riemann sphere” to make this exception make sense. In what is known as a stereographic projection, each point on the sphere is projected to a position on the plane, by drawing a line from the north pole, through the point on the sphere, and seeing where it intersects the plane. You can see a moving point in the stereographic projection:

But now, where will the north pole itself map to on the plane? Points near it go incredibly (infinitely) far out in whatever direction they are from the north pole, but the north pole itself becomes simply infinite without direction. It is both +infinite and -infinite (as well as +i infinite and -i infinite, and any other direction you might choose). By building Riemann surfaces on Riemann spheres, poles can be addressed, because the discontinuous shift from +infinite to -infinite is just the north pole.

Infinitely Infinite

What if we combine not a finite number of poles, but an infinite number of them? We'll take a look at what is sometimes today called the “essential singularity”, a pole of infinite order. We can create it with e1/z – if you work out the algebraic expansion, you'll see why. (The expansion is performed at the end of this video.) This video zooms in on the origin, showing the increasing density of infinites as we get closer to zero. The grid is present to help make the shape clearer, but keep in mind that it doesn't move as the image zooms in: